Theorem exmidontri2or 7428

Description: Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)

Assertion

Ref	Expression
exmidontri2or	|- (EXMID <-> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )

Distinct variable group:   x, y

Proof of Theorem exmidontri2or

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exmidontriim 7407	. . 3 |- (EXMID -> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
2		onelss 4478	. . . . . . . 8 |- ( y e. On -> ( x e. y -> x C_ y ) )
3	2	adantl 277	. . . . . . 7 |- ( ( x e. On /\ y e. On ) -> ( x e. y -> x C_ y ) )
4		orc 717	. . . . . . 7 |- ( x C_ y -> ( x C_ y \/ y C_ x ) )
5	3, 4	syl6 33	. . . . . 6 |- ( ( x e. On /\ y e. On ) -> ( x e. y -> ( x C_ y \/ y C_ x ) ) )
6		eqimss 3278	. . . . . . . 8 |- ( x = y -> x C_ y )
7	6, 4	syl 14	. . . . . . 7 |- ( x = y -> ( x C_ y \/ y C_ x ) )
8	7	a1i 9	. . . . . 6 |- ( ( x e. On /\ y e. On ) -> ( x = y -> ( x C_ y \/ y C_ x ) ) )
9		onelss 4478	. . . . . . . 8 |- ( x e. On -> ( y e. x -> y C_ x ) )
10	9	adantr 276	. . . . . . 7 |- ( ( x e. On /\ y e. On ) -> ( y e. x -> y C_ x ) )
11		olc 716	. . . . . . 7 |- ( y C_ x -> ( x C_ y \/ y C_ x ) )
12	10, 11	syl6 33	. . . . . 6 |- ( ( x e. On /\ y e. On ) -> ( y e. x -> ( x C_ y \/ y C_ x ) ) )
13	5, 8, 12	3jaod 1338	. . . . 5 |- ( ( x e. On /\ y e. On ) -> ( ( x e. y \/ x = y \/ y e. x ) -> ( x C_ y \/ y C_ x ) ) )
14	13	ralimdva 2597	. . . 4 |- ( x e. On -> ( A. y e. On ( x e. y \/ x = y \/ y e. x ) -> A. y e. On ( x C_ y \/ y C_ x ) ) )
15	14	ralimia 2591	. . 3 |- ( A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) -> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )
16	1, 15	syl 14	. 2 |- (EXMID -> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )
17		ontri2orexmidim 4664	. . . 4 |- ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> DECID z = { (/) } )
18	17	adantr 276	. . 3 |- ( ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) /\ z C_ { (/) } ) -> DECID z = { (/) } )
19	18	exmid1dc 4284	. 2 |- ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> EXMID )
20	16, 19	impbii 126	1 |- (EXMID <-> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   \/ w3o 1001   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   (/)c0 3491   {csn 3666  EXMIDwem 4278   Oncon0 4454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3889  df-tr 4183  df-exmid 4279  df-iord 4457  df-on 4459  df-suc 4462

This theorem is referenced by:  onntri52  7429